Showing papers on "Ring of integers published in 1998"
16 Aug 1998
TL;DR: This paper presents classes of algebraic interleavers that permute a sequence of bits with nearly the same statistical distribution as a randomly chosen interleaver when they are used in turbo-coding.
Abstract: In this paper we present classes of algebraic interleavers that permute a sequence of bits with nearly the same statistical distribution as a randomly chosen interleaver. When these interleavers are used in turbo-coding, they perform equal to or better than the average of a set of randomly chosen interleavers. They are based on a property of quadratic congruences over the ring of integers modulo powers of 2.
72 citations
01 Jan 1998
30 citations
TL;DR: In this paper, the authors established a criterion for verifying Bremner's conjecture for a given regular prime and used it to prove the conjecture for p⩽23,p≠17.
20 citations
TL;DR: In this article, the volume of the quotient of three-dimensional hyperbolic space by SL2 over the ring of integers of an imaginary quadratic number field was shown to be the same as that of Humbert.
Abstract: We close a gap in Humbert's classical calculation of the volume of the quotient of three-dimensional hyperbolic space by SL2 over the ring of integers of an imaginary quadratic number field.
10 citations
Posted Content•
TL;DR: In this article, it was shown that the local monodromy operator and its powers determine invariant cocycle classes under the decomposition group in the cohomology of the product $X \times X.
Abstract: In the paper we show that for a normal-crossings degeneration $Z$ over the ring of integers of a local field with $X$ as generic fibre, the local monodromy operator and its powers determine invariant cocycle classes under the decomposition group in the cohomology of the product $X \times X$. More precisely, they also define algebraic cycles on the special fibre of a resolution of $Z \times Z$. In the paper, we give an explicit description of these cycles for a degeneration with at worst triple points as singularities. These cycles explain geometrically the presence of poles on specific local factors of the L-function related to $X \times X$.
9 citations
TL;DR: An efficient algorithm for the approximation of real input signals, with arbitrarily small error, as elements of a quadratic number ring is provided and residual number system moduli restrictions for simplified multiplication within the ring are proved.
Abstract: Previous work has focused on performing residue computations that are quantized within a dense ring of integers in the real domain. The aims of this paper are to provide an efficient algorithm for the approximation of real input signals, with arbitrarily small error, as elements of a quadratic number ring and to prove residual number system moduli restrictions for simplified multiplication within the ring. The new approximation scheme can be used for implementation of real-valued transforms and their multidimensional generalizations.
8 citations
TL;DR: In this article, the authors studied the set of rational points of the group of components of an abelian variety over a discrete valuation field K. In particular, they looked at the cases where A_K is a Jacobian or a semiautomain variety having semi-stable reduction.
Abstract: Let A_K be an abelian variety over a discrete valuation field K. Let A be the Neron model of A_K over the ring of integers O_K of K and A_k its special fibre. We study the set of rational points of the group of components \phi_A of A_k. In particular, we look at the cases where A_K is a Jacobian or where A_K is an abelian variety having semi-stable reduction. We also consider algebraic tori over K instead of abelian varieties.
7 citations
TL;DR: In this article, a survey of solutions over the ring of integers in quadratic imaginary fields, where k > 0 squarefree, of the equation ax2 + by2 + cz2 = dxyz, where the coefficients are chosen as follows:
Abstract: The diophantine equation ax2+by2+cz2 = dxyz with a, b, c, d ∈ ℤ{0} and a, b, c ¦ d has been studied in connection with discrete subgroups of PSL(2, ℝ) ([R2, KR, K, Sch]). Rosenberger and Kern-Isberner have determined the complete set of integral solutions. Further Silverman ([S]) gave a description of the solutions of the equation x2 + y2 + z2 = dxyz, |d| ≥ 3, over orders in quadratic imaginary fields, whereas Bowditch, Maclachlan and Reid studied the equation x2 + y2 + z2 = xyz in order to describe the arithmetic once-punctered torus bundles ([BMR]). In this paper we give a survey of the set of solutions over the ring of integers Ok in quadratic imaginary fields, \(\mathbb{Q}\left( {\sqrt { - k} } \right)\), k > 0 squarefree, of the equation ax2 + by2 + cz2 = dxyz, where the coefficients are chosen as follows:
$$\bullet a,b,c,d \in O_k \backslash \{ 0\} ,\left. {a,b,c} \right|d,\left| {\tfrac{d} {{\sqrt {abc} }}} \right| \geqslant 3$$
$$\bullet a,b,c,d \in \mathbb{Z}\backslash \{ 0\} ,\left. {a,b,c} \right|d,1 \leqslant \left| {\tfrac{d} {{\sqrt {abc} }}} \right| < 3, resp. a = b = c = 1, d \in O_k \backslash \{ 0\} ,\left| d \right| < 3.$$
5 citations
TL;DR: In this article, it was shown that when L and K are division fields of a Lubin-Tate formal group, the ring of integers in L is a free rank one module over the associated order.
Abstract: Let L / K be extensions of the p-adic field
$ \Bbb Q _p $
. We show that when L and K are division fields of a Lubin-Tate formal group, then
$ \frak O _L $
, the ring of integers in L is a free rank one module over the associated order
$ \frak U _{L/K} $
. Chan and Lim had previously determined
$ \frak U _{L/K} $
without obtaining any structure results for
$ \frak O _L $
, except in the cyclotomic case. This extends, in the local case, previous results of Leopoldt, Chan and Lim, and Taylor.
5 citations
TL;DR: In this paper, some examples of locally unbounded topologizations of the field of rational numbers are given, and their completions are fields in which the rational number field is algebraically closed.
4 citations
TL;DR: The techniques of the association of long integers with polynomials for computations in the ring of integers are simplified and improved and applied to the computation of the signs of matrix determinants, Sturm sequences, and other algebraic and geometric predicates.
Journal Article•
TL;DR: In this article, Saito gave a formula analogous to that of Eichler for the difference of multiplicites of these characters in the trace of the representation of SL2(Fq )o nS2m(Γ(p)), where q = Np.
Abstract: Let S2m(Γ(p)) be the space of Hilbert modular cusp forms for the principal congruence subgroup with level p of SL2(OK) (here OK is the ring of integers of K, and p is a prime ideal of OK).Then we have the action of SL2(Fq )o nS2m(Γ(p)), where q = Np.When q is a power of an odd prime, for each SL2(Fq) we have two irreducible characters which have conjugate values mutually.In the case where K is the field of rationals, M.Eichler gives a formula for the difference of multiplicites of these characters in the trace of the representation of SL2(Fq )o nS2m(Γ(p)).In the case where K is a real quadratic field, H. Saito gives a formula analogous to that of Eichler for the difference.The purpose of this paper is to give a formula analogous to that of Eichler in the case where K is a totally real cubic field.
TL;DR: In this article, the general solution of three Diophantine equations in the ring of integers of the algebraic number field was given, which are related to the problem of determining the minimum distance in quasicrystals with fivefold symmetry.
Abstract: We give the general solution of three Diophantine equations in the ring of integers of the algebraic number field
$$Q\left[ {\sqrt 5 } \right]$$
. These equations are related to the problem of determining the minimum distance in quasicrystals with fivefold symmetry.
TL;DR: Theorem 5.1 as mentioned in this paper shows that the Birch-Swinnerton-Dyer conjecture for CM Q-curves holds up to a power of 2, which is the best known result.
Abstract: Let K ⊂ C be an imaginary quadratic field with prime discriminant −p < −3, ring of integers OK = O and class group ClK of (odd) order hK = h. The jinvariant j(O) generates a field F/Q of degree h such that H = FK is the Hilbert class field of K. Suppose A/F is a Q-curve with j-invariant j(O); thus, A is an elliptic curve which, over H, is isogenous to each of its Galois conjugates, and has complex multiplication by O. We let B = ResF/QA be the h-dimensional abelian variety over Q obtained from A by restriction of scalars. Any two such A (and any two such B) are quadratic twists of one another: letting A(p) denote the canonical curve of discriminant ideal (−p3), with restriction B(p), we have A = A(p) (and B = B(p)) for some quadratic discriminant D. We refer the reader to Gross [Gr1] for general facts about CM Q-curves. Much progress has been made recently on proving the conjecture of Birch and Swinnerton-Dyer for these curves. For instance, if L(1, A/F ) = 0, and h = 1, Rubin [Ru2] has proved that the Birch-Swinnerton–Dyer conjecture for A/F holds up to a power of 2. Note that the ring R+ of Q-endomorphisms of B (or B(p)) is an order in T = R+ ⊗ Q, a totally real field of degree h over Q. The Tate-Shafarevich group ∐∐ B/Q is a finite module over R+; our main goal in this paper is to gain insight into the structure of this module via L-series in some special cases. Namely, suppose p ≡ 3 mod 8 and A = A(p)−3. Also, assume that R+ is integrally closed. Let ψ be a Hecke character of K such that ψ ◦ NH/K is the Hecke character attached to A/H. This choice of ψ gives rise to an embedding of T in R (see section 2). We define an algebraic integer s = 0 in FT as a sum of certain modified elliptic units first introduced by Gross [Gr3] and show that there is a (unique) integral ideal f of R+ whose lift to FT is generated by s. Our starting point is a formula (Theorem 2) expressing L(1, ψ) as a period times s, showing, in particular, that this central critical value does not vanish. Writing L(s, A/F ) as a product of Hecke L-series, calculating the local factors in the Birch-Swinnerton–Dyer conjecture, and applying our formula together with results of Coates, Wiles, Arthaud, and Rubin [CW], [Ar], [Ru1], we obtain Main Theorem(Theorem 5) With the above assumptions and notation, A(F ) = B(Q) is finite. If the Birch-Swinnerton–Dyer conjecture holds for A/F (or for
Patent•
13 Feb 1998
TL;DR: In this article, the authors proposed a scheme to obtain the strength equivalent to or higher than heretofore to complete cryptoanalysis and to enable the stronger encryption to a broadcast attack by forming the two prime ideals at the integer ring on an algebraic field respectively as secret keys and forming the width of these two prime ideal ideals as public keys.
Abstract: PROBLEM TO BE SOLVED: To obtain the strength equivalent to or higher than heretofore to complete cryptoanalysis and to enable the stronger encryption to a broadcast attack by forming the two prime ideals at the integer ring on an algebraic field respectively as secret keys and forming the width of these two prime ideals as public keys. SOLUTION: This apparatus has a key formation processing section 1, a signal communication device 2 and a signal reception device 3. The width of the two prime ideals P, Q in the integer ring O of the arbitrary algebraic field is defined as N. At this time, the determination of α from the power α (k is a natural number of >=2) to a modulus N of the arbitrary element α of O is equivalent to the feasibility of the prime ideal analysis of N. If the prime ideal analysis of N is possible, α is determined by determining the greatest common measure of the polynominal on the finite field. N is, thereupon, placed as the public key, P, Q are placed as the secret keys, respectively and plaintext αis enencrypted by α modN, by which ciphertext is obtd. The ciphertext is decrypted by determining the solution of the polynomial on the finite field, by which the plaintext α is obtd.
Posted Content•
TL;DR: In this article, the following conjecture of Kitaoka was shown to be false: if a finite subgroup is invariant under the action of a Galois extension, then it is contained in a finite abelian subgroup.
Abstract: We discuss the following conjecture of Kitaoka: if a finite subgroup $G$ of $GL_{n}(O_{K})$ is invariant under the action of $Gal(K/\Bbb Q)$ then it is contained in $GL_{n}(K^{ab})$. Here $O_{K}$ is the ring of integers in a finite, Galois extension $K$ of $\Bbb Q$ and $K^{ab}$ is the maximal, abelian subextension of $K$. Our main result reduces this conjecture to a special case of elementary abelian $p-$groups $G$. Also, we construct some new examples which negatively answer a question of Kitaoka.
TL;DR: In this article, the construction of non-decomposable Hermitian forms over the ring of integers of an imaginary quadratic field has been studied for the first time.
Abstract: Methods are presented for the construction of non-decomposable Hermitian forms over the ring of integers of an imaginary quadratic field.
TL;DR: In this article, the authors give the upper bound of the number of cycles in a strongly connected digraph by using the theory of commutative rings and show that if the digraph is strongly connected, then there is no Hamilton cycle.
Posted Content•
TL;DR: In this article, a finite spanning set for the highest nonvanishing rational cohomology group of a subgroup of a number field with ring of integers O was constructed from a classical construction from the geometry of numbers and the theory of modular symbols.
Abstract: Let K be a number field with ring of integers O, and let G be a finite-index subgroup of SL(n,O). Using a classical construction from the geometry of numbers and the theory of modular symbols, we exhibit a finite spanning set for the highest nonvanishing rational cohomology group of G.
01 Jan 1998
TL;DR: In this article, the Eisenstein series of general cofinite groups was studied in detail for the group PSL(2, Thuong) ⊂ PSL (2, ℂ).
Abstract: Suppose that \(K = Q\left( {\sqrt D } \right) \) is an imaginary quadratic number field (D ∈ ℤ, D < 0 and square-free) and let Thuong be the ring of integers of K. We consider here the group PSL(2, Thuong) ⊂ PSL(2, ℂ). We already know from Chapter 7 that Γ is a discrete subgroup which is cofinite but not cocompact. We study the Eisenstein series defined in Chapter 3 in detail for the group PSL(2, Thuong). In fact we shall establish most of the general facts proved in Section 6.1 for the Eisenstein series of general cofinite groups by direct number theoretic methods. We shall for example relate the determinant of the scattering matrix to the zeta function of the Hilbert class field of K. The control we have over the Eisenstein series will also in turn imply many interesting number theoretic results.
TL;DR: In this paper, an FPGA implementation of Frequency Sampling Filtering (FSF) is presented which offers an 86% complexity reduction compared with standard non-recursive FIR designs.
Abstract: Frequency sampling filters (FSFs) are of interest to designers of filter banks due, in part, to their intrinsic low complexity and linear phase behavior. FSF designs rely on exact pole-zero annihilation and are often found in embedded applications. Exact FSF pole-zero annihilation can be guaranteed by using polynomial filters defined over an integer ring in the residue number system (RNS) along with algebraic integers. An FPGA implementation example is presented which offers an 86% complexity reduction compared with standard nonrecursive FIR designs.
TL;DR: In this paper, it was shown that the Bass-cyclic units and bicyclic units generate a subgroup of finite index in the group of units of OG, where O is the ring of integers in the Brauer field Q(ζ n ).
Abstract: Let G be a finite group of order n. It is known that the Bass-cyclic units and bicyclic units generate a subgroup of finite index in the group of units of OG, where O is the ring of integers in the Brauer field Q(ζ n ).We now prove a finite index theorem for the Solomon field Q(ζ2k ), where k= π p∣n p.
TL;DR: In particular, the authors gave an explicit expression for the inversion factor (α / β ) l ǫ( β / α ) −1 l of the l th power residue symbol over the cyclotomic field of l th roots of unity.
21 Jun 1998
TL;DR: An account of some largely experimental results about the successive minima of the ring of integers of an algebraic number field endowed with its canonical Euclidean norm throws light on some interesting facts which could be important for the algorithmic theory of number fields.
Abstract: We give an account of some largely experimental results about the successive minima of the ring of integers of an algebraic number field endowed with its canonical Euclidean norm. This throws light on some interesting facts which could be important for the algorithmic theory of number fields.